Math 2.0

Integral equations Singular integral equations Example

Carleman's singular integral equation

Builds on Cauchy principal valueFinite Hilbert transformRiemann–Hilbert problemSokhotski–Plemelj theoremSingular integral equationAirfoil equation

Example

Let aa and bb be real constants with b≠0b \ne 0, and let ff be a given function on the interval (−1,1)(-1, 1). Carleman's equation asks for a function φ\varphi on (−1,1)(-1, 1) with a φ(x)+bπ PV ⁣∫−11φ(t)t−x dt=f(x),−1<x<1,a\, \varphi(x) + \frac{b}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x), \quad -1 < x < 1, where PV\mathrm{PV} is the Cauchy principal value. It is a singular integral equation of the second kind with constant coefficients on an arc, and its singular part is the finite Hilbert transform.

A check with an explicit solution uses two classical principal value integrals, valid for −1<x<1-1 < x < 1: 1π PV ⁣∫−111−t2t−x dt=−x,1π PV ⁣∫−11dt1−t2 (t−x)=0.\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{t - x}\, dt = -x, \qquad \frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{dt}{\sqrt{1 - t^2}\,(t - x)} = 0. Hence φ(t)=1−t2\varphi(t) = \sqrt{1 - t^2} solves the equation with right side f(x)=a1−x2−bxf(x) = a\sqrt{1 - x^2} - b x. The function (1−t2)−1/2(1 - t^2)^{-1/2} solves it with right side f(x)=a(1−x2)−1/2f(x) = a (1 - x^2)^{-1/2}.

Remarks

Put Φ(z)=12πi∫−11φ(t)t−z dt\Phi(z) = \frac{1}{2\pi i} \int_{-1}^{1} \frac{\varphi(t)}{t - z}\, dt for z∉[−1,1]z \notin [-1, 1]. By the Sokhotski–Plemelj theorem, Φ+−Φ−=φ\Phi^+ - \Phi^- = \varphi and Φ++Φ−=1πi PV ⁣∫−11φ(t)t−x dt\Phi^+ + \Phi^- = \frac{1}{\pi i}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt on the cut. The equation becomes (a+ib) Φ+=(a−ib) Φ−+f(a + ib)\, \Phi^+ = (a - ib)\, \Phi^- + f, a Riemann–Hilbert problem on the segment with constant coefficient (a−ib)/(a+ib)(a - ib)/(a + ib) of modulus 11. Solving it gives the solutions in closed form, with powers of 1−x1 - x and 1+x1 + x at the endpoints whose exponents are fixed by the argument of a+iba + ib. For a=0a = 0 the equation is the airfoil equation.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Cauchy principal value, concept, no evidence levelCauchy principal valueconceptComplex numbers, object, no evidence levelComplex numbersobjectElement, definition, no evidence levelElementdefinitionFinite Hilbert transform, concept, no evidence levelFinite Hilbert transf…conceptFunction, concept, no evidence levelFunctionconceptHolomorphic function, concept, no evidence levelHolomorphic functionconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSingular integral equation, concept, no evidence levelSingular integral equ…conceptContour integral, definition, no evidence levelContour integraldefinitionSet, definition, no evidence levelSetdefinitionAirfoil equation, example, no evidence levelAirfoil equationexampleLaplace transform, definition, no evidence levelLaplace transformdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionCauchy principal value, definition, no evidence levelCauchy principal valuedefinitionWinding number, definition, no evidence levelWinding numberdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionHeaviside step function, object, no evidence levelHeaviside step functi…objectExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionDirac delta function, object, no evidence levelDirac delta functionobjectSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionHölder continuity, definition, no evidence levelHölder continuitydefinitionSokhotski–Plemelj theorem, statement, no evidence levelSokhotski–Plemelj the…statementRiemann–Hilbert problem, problem, no evidence levelRiemann–Hilbert probl…problemCarleman's singular integral equation, example, no evidence levelCarleman's singular i…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy principal value concept
  2. Complex numbers object
  3. Element definition
  4. Finite Hilbert transform concept
  5. Function concept
  6. Holomorphic function concept
  7. Proposition concept
  8. Real numbers object
  9. Singular integral equation concept
  10. Contour integral definition
  11. Set definition
  12. Airfoil equation example
  13. Laplace transform definition
  14. Implication definition
  15. Predicate definition
  16. Cauchy principal value definition
  17. Winding number definition
  18. Ordered pair definition
  19. Heaviside step function object
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Dirac delta function object
  24. Subset definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Function definition
  28. Metric space definition
  29. Lipschitz map definition
  30. Hölder continuity definition
  31. Sokhotski–Plemelj theorem statement
  32. Riemann–Hilbert problem problem
  33. Carleman's singular integral equation example

Further reading

  1. Web https://en.wikipedia.org/wiki/Singular_integral_operators_on_closed_curves
About this page
TagSPV5SRAS9H
URIm2://SPV5SRAS9H
Kindexample
WritingW0 stub
MSC45E05, 45E99
Instance ofSingular integral equation
UsesCauchy principal value, Finite Hilbert transform, Riemann–Hilbert problem, Sokhotski–Plemelj theorem, Singular integral equation, Airfoil equation

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